From (6) we see that:
(9)
( nι
Z = ∑
V i=1
1-1 ъ ■ - A σ-1
x≈1iσ +∑ x2Γ
i =1 )
Making a substitution of (7) and (8) into (9) leaves us with:
(10)
Z = xrnσ ∑
1 11
j V Pj 1 )
Collecting the terms in (10) that relate to input xi 11 and simplifying leads to:
(11)
Z = xi11 Pn ∑ P j1σ
V j
(_
+ xi11 Pil Σ (Pj2 t)
V j
1-σ
)
σ
Aσ-1
This can be written as:
(12) Z = xn1 Ph P - -
1
( n1 n2 A 1-n
, where p = I ∑ p}1 σ +∑(pj21)1-σ I
Due to the symmetric fashion in which the
V i =1 i =1 )
producer service varieties enter the sub-production function of the manufacturing firm,
this expression of P can be simplified to read:
(13) p = (n 1 p 1-σ + n2( p21)1-σ )1-σ
, which the same as Equation (2). To see that P is the cost function, we solve for input
demand:
(14) xfl1 = ZPi-σ P p
Letting the cost function be evaluated at a production of Z=1 leads to:
√-σ
(15) c = ∑ P11-σPσ +∑ (Pi21)1-σ Pσ = Pσ (n 1 p1-σ + n2(P21)1-σ )1-σ = PσP1-σ = P
i i
23
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